Using Warrants As a Window to Epistemic Framing
Thomas J. Bing and Edward F. Redish
Department of Physics, University of Maryland, College Park, MD 20742, USA
Abstract. Mathematics can serve many functions in physics. It can provide a computational system, reflect a physical
idea, conveniently encode a rule, and so forth. A physics student thus has many different options for using mathematics
in his physics problem solving. We present a short example from the problem solving work of upper level physics students and use it to illustrate the epistemic framing process: “framing” because these students are focusing on a subset of
their total math knowledge, “epistemic” because their choice of subset relates to what they see (at that particular time) as
the nature of the math knowledge in play. We illustrate how looking for students’ warrants, the often unspoken reasons
they think their evidence supports their mathematical claims, serves as a window to their epistemic framing. These warrants provide a powerful, concise piece of evidence of these students’ epistemic framing.
Keywords: Epistemic Resources, Upper Level Physics, Problem Solving
PACS: 01.40.Fk, 01.30.lb
INTRODUCTION
Mathematics is central to learning physics. As
physics students progress to higher and higher levels
of physics classes, the use of mathematics becomes
evermore complex. Certainly the mathematics becomes more complex in a conceptual sense—upper
level physics uses increasingly technical math. This
paper, however, focuses on a different aspect of the
complexity of mathematics in physics classes: epistemic complexity. Math in physics can have many
different natures: it can be a computational scheme,
reflect a physical idea, conveniently encode a rule, and
so forth.
This paper will use the idea of epistemic framing to
model how physics students juggle these various natures of their mathematics. We will examine the warrants these students use in their arguments as a primary
source of evidence for their epistemic framing.
EPISTEMIC FRAMING
We turn first to a brief overview of two cognitive
modeling ideas: epistemic resources and framing.
These ideas will be applied to a sample episode of
students’ work later in the paper.
Epistemic Resources
Epistemic resources deal with how students perceive the nature of the knowledge under current con-
Bing and Redish
sideration. Do they see scientific knowledge as fixed
and absolute or as being relative to one’s point of
view? Do they view scientific knowledge as something they can construct for themselves or as something handed down from an authority figure [1]?
Epistemic resources activate and deactivate, leading a student to adopt a certain stance towards the
physics knowledge at hand. Epistemic stances are
highly sensitive to context [2]. A student working on a
physics problem might spend time trying to build a
story explaining why a certain result happened (hence
treating knowledge as personally constructed) but
might then turn around and search a textbook for a rule
pertaining to his question (hence treating knowledge as
handed down from authority). There are detailed published examples of such in-the-moment shifts in students’ reasoning [3,4].
Framing
Framing is the (often subconscious) choice the
mind makes regarding “What kind of activity is going
on here?” [5]. A person cannot actively consider all
her knowledge in a given situation. Some subset is
selected based on the mind’s (often very quick) interpretation of what kind of situation is at hand.
Framing has been studied in a wide array of academic disciplines including linguistics, sociology, art,
psychology, and anthropology. All of these studies
implicitly agree on the existence of what has been
called “Felicity’s Condition” [6]. Felicity’s Condition
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is the unspoken premise naturally adopted by an individual that incoming information, whether it be spoken, read, observed, etc, comes from a rational source,
and it is thus up to the individual to attempt to contextualize and hence interpret that incoming information.
Framing is the process by which the mind attempts this
contextualization and interpretation.
That is not to say this contextualization always
happens flawlessly. Different individuals can certainly
frame the same incoming information in different
ways [5]. Miscommunications arise when two individuals frame their interaction differently, each bringing a different subset of their available resources to
bear on the situation. This paper presents such an example.
How a person frames a situation has many ramifications: social, conceptual, emotional, etc. This paper
will focus on the epistemic ramifications of students’
framing. What do they presently see as the nature of
the mathematics knowledge they are using? That is,
what epistemic resources have been tapped by their
current framing of their problem solving?
But how can one best identify and analyze what
“epistemic resources have been tapped” by physics
students? We suggest: look for the (sometimes implicit) warrants they use in their mathematical arguments.
ARGUMENTATION
Argumentation theory is a field of research about
how people build justifications and communicate them
effectively to each other. Since argumentation theory
will provide a lens (i.e. the idea of warrants) for observing students’ epistemic framing, this paper briefly
turns to an overview of several threads of argumentation research.
There are several subfields that are sometimes colloquially lumped under “argumentation theory” [7].
On one end of the continuum is what is best called
formal logic. Studies in formal logic deal with relatively clean and straightforward methods of proofmaking that can easily be decontextualized from whatever given situation is at hand. Formal logic chains
like “if A then B, if B then C but not D, etc.” are common to such studies.
A second branch of research [7], the one that is
most often actually called “argumentation theory”,
includes what is often called rhetoric. This field of
research focuses most on presenting, as opposed to
having, an argument. It attempts to parse the content
of a given argument into some kind of structure and
often carries some sort of evaluative tone with regard
to that structure. A central pillar of this field, and an
important basis for this paper’s analysis, is the work of
Bing and Redish
Stephen Toulmin [8]. He devised an often-cited system for parsing an argument into such parts as claims,
data, and warrants. A person will make a statement,
the claim, that requires proof. They will then offer one
or more relevant facts, the data. The warrant is the
bridge, sometimes unspoken, that explains how the
given data relates to the claim at hand. For example, I
might state that Jack Nicklaus is the greatest golfer
alive (claim) because he won the Masters six times
(data). The relevant warrant that would link this data
to that claim would be that the Masters is a very prestigious tournament that is played every year on Augusta National, one of the most difficult courses on the
planet.
Many naturally occurring arguments are more
nebulous in structure than an argument fitting a clean
Toulmin structure. Justifications that are logically
unsound are often treated as acceptable in informal,
real-time situations. A third branch of research, often
gathered under the label discourse analysis [7], concerns itself primarily with the in-the-moment patterns
people employ in their speech and thought as they
construct and communicate arguments.
These in-the-moment argument constructions are
often verbally incomplete. They often refer to a body
of knowledge that the speaker (correctly or incorrectly) assumes he shares with the listener. These
flow-of-conversation arguments sometimes have holes
in them that are consciously or unconsciously overlooked.
This paper’s analysis incorporates elements from
what van Rees calls “argumentation theory” and “discourse analysis”. Toulmin’s idea of a warrant is central. Tracking the warrants physics students use in
their mathematical arguments provides an excellent
window onto their epistemic framing. We are not
concerned, however, with evaluating the students’
arguments. We are concerned with looking closely at
a few “in-the-moment patterns” they employ in their
speech and how they relate to the students’ epistemic
framing of their math use in physics.
EXAMPLE
This example comes from a video of two students,
a sophomore (S1) and a junior (S2), working outside
of class on a homework problem from a Mathematical
Methods in Physics class. Tracking their (shifting)
warrants will give access to the epistemic framing dynamic that is central to their conversation.
Their homework question asks them to use
B r
r
W = ∫ F ⋅ dr to calculate the work done by the gravitaA
tional force from a mass, M, at the origin on a smaller
mass, m, that moves from (1,1) to (3,3) along two dif-
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ferent paths. The first path goes from (1,1) to (3,3)
directly while the second goes horizontally to (3,1) and
then vertically to (3,3). They have decided to ignore
the GMm constants and have written
3 2
3
3
(1)
1
1
1
∫
2
r2
dr = ∫
1
y2 + 9
dy + ∫
1
x2 +1
dx
While the integral on the left for the radial path is correct within a minus sign, the integrals for the other
path are not. There is a missing cosine term from the
dot product definition of work. The students have also
drawn a figure on the board that shows the two paths.
FIGURE 1: The students’ two-path diagram.
The students discuss their expected result:
1. S1: What’s the problem?
2.
You should get a different answer
3.
from here for this. (Points to Fig.1)
4. S2: No, no, no.
5. S1: They should be equal?
6. S2: They should be equal.
7. S1: Why should they be equal?
8.
This path is longer if you think about it.
(Points to Fig.1)
9. S2: Because force, err,
10. because work is path independent.
11. S1: This path is longer, so it should have,
12.
this number should be bigger than
(Points to Fig.1)
13. S2: Work is path independent. If you
14.
go from point A to point B, doesn’t matter
15.
how you get there, it should take
16.
the same amount of work.
17. S1: OK, that’s assuming Pythagorean
18.
Theorem and everything else add[s].
19.
Well, OK, well is this—what was the
20.
answer to this right here? (Points to Eq.1)
21.
What was that answer?
22. S2: Yeah, solve each integral numerically.
23. S1: Yeah, what was that answer?
(They turn to a computer with Mathematica)
Lines 1 to 6 introduce the crux of the episode. S1
thinks there should be different amounts of work done
on the small mass along the two different paths. S2
believes the work done should be the same.
S1 offers a justification for his claim in lines 7 and
8 when he challenges S2’s same-work assertion. “This
path is longer if you think about it.” The mathematical
Bing and Redish
r r
W A→ B = ∫ F ⋅ dr , is essentially
B
definition of work,
A
“force times distance”. Since the two-part path from
A to B is physically longer than the direct route, it
seems to follow that more work is done along the
longer path.
In formal argumentation theory language [8], S1 is
“claiming” that more work is done along the two-part
path, and he offers the “data” that the two-part path is
longer. There is an unspoken “warrant” that connects
his data to his claim: mathematics should align with
the physical systems under study in a physics class.
The goodness-of-fit between the math at hand and the
physical system attests to the validity of one’s conclusions. The work formula seems to say “force times
distance” to S1. The two-part path has more “distance”, and S1 thus draws justification for his answer.
Students’ warrants offer excellent evidence for
their epistemic framing of their math use. S1’s warrant thus suggests he is framing his activity as Physical
Mapping. S1 has currently activated epistemic resources that view mathematics knowledge as encoding
physical ideas.
S2 not only has a different answer than S1, but he
is also framing his use of mathematics in a different
way. S2 claims that the work done on the small mass
should be the same along the two paths “because work
is path independent” (lines 9 and 10). His data is the
(incompletely recalled) theorem: “work is path independent”. The unspoken warrant that S2 is relying on
concerns the common use of rules and definitions in
math and physics: sometimes previous results are
simply taken as givens for speed and convenience. S2
is framing their math use as Invoking Authority. He is
thinking with epistemic resources that see mathematics
as a system for packaging and quoting rules and previous results. Such a framing is not necessarily naïve.
Even expert physicists don’t necessarily start every
argument from absolute first principles every time.
After hearing S2’s counterargument, S1 repeats
himself. In lines 11 and 12, he restates his longer-path
justification and again points to the relevant features of
the diagram they had previously drawn on the board.
S2 responds by restating “work is path independent” in
line 13 and again, in a slightly different way, in lines
14 to 16.
The most important observation in this clip is that
S1 and S2 are disagreeing over much more than the
answer itself. Explicitly, they are debating whether or
not more work is done along the longer path. Implicitly, they are arguing over the most useful way to
frame their present use of mathematics. S1 never explicitly says, “Please respond to my claim in a way
that maps our math to some detail of the physical
situation I may have overlooked”. His phrasing and
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gesturing in his initial argument (lines 7 and 8) implies
this framing request, though.
When S2 responds with his rule citation, he is not
merely arguing for a different answer. He is pushing
for a different type of warrant for judging the validity
of a given answer. S2’s Invoking Authority framing
may have even prevented him from really hearing
what S1 was saying. S1’s framing request may have
passed by S2 unnoticed because he was too caught up
in the subset of all his math resources that his Invoking
Authority framing had activated within his mind. At
any rate, S2 responds in lines 9 and 10 with a different
type of justification than what S1 was expecting.
When S1 repeats himself in lines 11 and 12, he is
implicitly repeating his bid for a Physical Mapping
framing. One can imagine a situation when S2’s Invoking Authority justification would simply be accepted without incident, but here it did not align with
S1’s present framing. S2 does not respond to this reframing request and repeats his answer as he remains
with his Invoking Authority framing.
There is thus an intense framing argument going on
under the surface of this debate. Sensing that he is not
making any headway in the framing battle, S1 now
moves to shift both himself and S2 into a third framing.
S1 moves to reframe the discussion in lines 19 to
21. He points to the integrals they’ve written and asks,
“Well, OK…what was the answer to this right here?
What was that answer?” He is calling for someone to
evaluate each of their expressions for the work so that
he can compare the numeric results. This argument
relies on another kind of warrant. Mathematics provides one with a standardized, self-consistent set of
manipulations. Performing a computation, or having a
computer do it for you, according to these rules will
give a valid, trustable result. S1 is moving to reframe
their math use as Calculation. Epistemic resources
associated with this framing see mathematics as a system of algorithmic processes.
Even though S1 doesn’t explicitly detail the new
warrant he is proposing, S2 is quick to tune into it. He
immediately responds, “Yeah, solve each integral numerically” (line 22). Compare this successful epistemic frame negotiation with the struggle earlier in this
snippet. Lines 1 to 16 had S1 pushing for Physical
Mapping while S2 lobbied for Invoking Authority.
Both stuck to their positions, resulting in an inefficient
conversation. Neither was accepting what the other
was trying to say. Lines 19 to 22 have S1 and S2
agreeing, for the moment, on what type of mathematical justification should count.
Bing and Redish
CLOSING DISCUSSION
The sample student conversation given above contains a series of epistemic framing pushes and pulls.
They work to find a common ground, a shared perspective of what type of argument would be appropriate.
We do not claim that the three epistemic framings
named in this episode represent rigidly compartmentalized modes of thought. We have often observed
mixings, like a student using a calculation scheme to
back up a physical intuition. Our main point here has
been to illustrate how tracking the (sometimes implicit) warrants students use is an excellent way to
gather evidence of their epistemic framing of their
math use. Epistemic framing issues are often a powerful (if sometimes implicit) driving force behind students’ speech and thought.
ACKNOWLEDGMENTS
The authors wish to thank the Physics Education
Research Group at the University of Maryland for
their important contributions. This work was supported by NSF grants DUE 05-24987 and REC 0440113 and a Graduate Research Fellowship.
REFERENCES
1. D. Hammer and A. Elby, “On the Form of a Personal
Epistemology,” in Personal Epistemology: The Psychology of Beliefs about Knowledge and Knowing, edited
by B. K. Hofer and P. R. Pintrich, Mahwah, NJ: Erlbaum, 2002, pp. 169-190.
2. D. Hammer, Cognition and Instruction 12(2), 151-183
(1994).
3. D. Hammer, “The Variability of Student Reasoning,
Lecture 2: Transitions,” in Proceedings of the International School of Physics, “Enrico Fermi”, Course CLVI,
edited by E. F. Redish and M. Vincentini, Bologna, Italy:
Societa Italiana di Fisica, 2004, pp. 301-319.
4. D. Hammer, A. Elby, R. E. Scherr, and E. F. Redish,
“Resources, Framing, and Transfer,” in Transfer of
Learning: From a Modern Multidisciplinary Perspective, edited by J. P. Mestre, Greenwich, CT: Information
Age, 2005, pp. 89-119.
5. D. Tannen, “What’s in a Frame? Surface Evidence for
Underlying Expectations,” in Framing in Discourse, edited by D. Tannen, New York: Oxford University Press,
1993, pp. 14-56.
6. E. Goffman, “Frame Analysis of Talk,” in The Goffman
Reader, edited by C. Lemert and A. Branaman, Malden,
MA: Blackwell Publishing Limited, 1997, pp. 167-200.
7. M. A. van Rees, Journal of Pragmatics, 39(8), 14541463 (2007).
8. S. Toulmin, The Uses of Argument, Cambridge, UK:
Cambridge University Press, 1958.
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